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Properties

Label 16.0.14978832121...5625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{6}\cdot 97^{4}\cdot 101^{8}$
Root discriminant $57.67$
Ramified primes $5, 97, 101$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group 16T1275

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21793, 17411, -3747, 8281, 15452, -1102, -913, 3343, 1639, -550, 299, 228, 79, -11, 22, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 22*x^14 - 11*x^13 + 79*x^12 + 228*x^11 + 299*x^10 - 550*x^9 + 1639*x^8 + 3343*x^7 - 913*x^6 - 1102*x^5 + 15452*x^4 + 8281*x^3 - 3747*x^2 + 17411*x + 21793)
 
gp: K = bnfinit(x^16 + 22*x^14 - 11*x^13 + 79*x^12 + 228*x^11 + 299*x^10 - 550*x^9 + 1639*x^8 + 3343*x^7 - 913*x^6 - 1102*x^5 + 15452*x^4 + 8281*x^3 - 3747*x^2 + 17411*x + 21793, 1)
 

Normalized defining polynomial

\( x^{16} + 22 x^{14} - 11 x^{13} + 79 x^{12} + 228 x^{11} + 299 x^{10} - 550 x^{9} + 1639 x^{8} + 3343 x^{7} - 913 x^{6} - 1102 x^{5} + 15452 x^{4} + 8281 x^{3} - 3747 x^{2} + 17411 x + 21793 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14978832121137903853657515625=5^{6}\cdot 97^{4}\cdot 101^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.67$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 97, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{8} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{25} a^{12} - \frac{1}{25} a^{11} - \frac{2}{25} a^{9} + \frac{4}{25} a^{8} + \frac{6}{25} a^{7} - \frac{11}{25} a^{6} - \frac{11}{25} a^{5} - \frac{11}{25} a^{3} - \frac{12}{25} a^{2} + \frac{9}{25} a + \frac{8}{25}$, $\frac{1}{25} a^{13} - \frac{1}{25} a^{11} - \frac{2}{25} a^{10} + \frac{2}{25} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{3}{25} a^{6} - \frac{11}{25} a^{5} - \frac{11}{25} a^{4} + \frac{2}{25} a^{3} - \frac{3}{25} a^{2} - \frac{8}{25} a + \frac{8}{25}$, $\frac{1}{25} a^{14} + \frac{2}{25} a^{11} + \frac{2}{25} a^{10} - \frac{2}{25} a^{9} + \frac{9}{25} a^{8} + \frac{4}{25} a^{7} + \frac{8}{25} a^{6} + \frac{8}{25} a^{5} - \frac{3}{25} a^{4} + \frac{6}{25} a^{3} + \frac{2}{5} a^{2} + \frac{2}{25} a - \frac{7}{25}$, $\frac{1}{15205243219588442395063241225} a^{15} + \frac{53401729477521135850787113}{3041048643917688479012648245} a^{14} + \frac{143269283897626963205588178}{15205243219588442395063241225} a^{13} - \frac{134861224875082529565839473}{15205243219588442395063241225} a^{12} + \frac{2053066348691436536786902}{323515813182732816916239175} a^{11} + \frac{405188580815744353015205767}{15205243219588442395063241225} a^{10} + \frac{3290665437578355103284749}{3041048643917688479012648245} a^{9} + \frac{3274883685843875884862810494}{15205243219588442395063241225} a^{8} - \frac{4811868096704522594402910122}{15205243219588442395063241225} a^{7} + \frac{1689821777459370116043094592}{15205243219588442395063241225} a^{6} + \frac{1774961204951045365067928484}{15205243219588442395063241225} a^{5} + \frac{484463963308162540682789288}{15205243219588442395063241225} a^{4} - \frac{2067014413695584532613339039}{15205243219588442395063241225} a^{3} + \frac{6640498933036148030663238553}{15205243219588442395063241225} a^{2} + \frac{6761299658811717059674155819}{15205243219588442395063241225} a - \frac{6615973642704284270143812476}{15205243219588442395063241225}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{12}$, which has order $12$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4906436.08499 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1275:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1275
Character table for t16n1275 is not computed

Intermediate fields

\(\Q(\sqrt{101}) \), 4.4.4947485.1, 8.0.122388039126125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
101Data not computed